"""
DaoMath Core Axioms (法则数学核心公理)
基于 ±0 先天均衡的公理化体系
"""

from typing import TypeVar, Generic, Callable, Optional, Any
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
import numpy as np

# ============================================================================
# A1: 先天均衡公理 (Innate Balance Axiom)
# ============================================================================

@dataclass(frozen=True)
class InnateBalance:
    """
    A1: ±0 = {+1, -1, 0_innate}
    
    先天均衡态：三元对称的原初状态
    """
    positive: complex = 1.0
    negative: complex = -1.0
    innate_zero: complex = 0.0
    
    def verify(self) -> bool:
        """验证均衡性质"""
        return (
            abs(self.positive + self.negative + self.innate_zero) < 1e-10 and
            abs(self.positive * self.negative + 1) < 1e-10  # (+1)(-1) = -1
        )
    
    def to_array(self) -> np.ndarray:
        """转换为数组表示"""
        return np.array([self.positive, self.negative, self.innate_zero])
    
    def __repr__(self) -> str:
        return f"±0 = {{{self.positive}, {self.negative}, {self.innate_zero}}}"


class BalancePreservingMap(ABC):
    """保持均衡性的映射（同态）"""
    
    @abstractmethod
    def __call__(self, balance: InnateBalance) -> InnateBalance:
        pass
    
    @abstractmethod
    def preserves_sum(self, b: InnateBalance) -> bool:
        """验证: f(+) + f(-) + f(0) = 0"""
        pass


# ============================================================================
# A2: 对偶互生公理 (Duality Genesis Axiom)
# ============================================================================

T = TypeVar('T')

@dataclass(frozen=True)
class DualPair(Generic[T]):
    """
    A2: ⊕/⊖ 对偶对
    
    任意概念A必有其对偶¬A，且二者共生互依
    """
    positive: T
    negative: T
    
    def verify_duality(self, metric: Callable[[T, T], float]) -> bool:
        """验证对偶关系: d(A, ¬A) = 最大距离"""
        return metric(self.positive, self.negative) > 0
    
    def annihilate(self) -> InnateBalance:
        """湮灭回归 ±0"""
        return InnateBalance()


# ============================================================================
# 阴阳对 (YinYangPair) - 对偶的物理实现
# ============================================================================

@dataclass
class YinYangPair:
    """
    阴阳对：对偶概念的动力学实现
    
    阴(Yin)与阳(Yang)相互转化，保持总量守恒
    """
    yin: float  # 阴性分量
    yang: float  # 阳性分量
    coupling: float = 1.0  # 耦合强度
    
    def __post_init__(self):
        """验证初始均衡"""
        if not self.is_balanced():
            raise ValueError(f"Initial state not balanced: yin={self.yin}, yang={self.yang}")
    
    def is_balanced(self, tolerance: float = 1e-6) -> bool:
        """验证阴阳守恒: yin + yang = const"""
        return abs(self.total() - (self.yin + self.yang)) < tolerance
    
    def total(self) -> float:
        """总量（守恒量）"""
        return self.yin + self.yang
    
    def imbalance(self) -> float:
        """不平衡度：yang - yin"""
        return self.yang - self.yin
    
    def evolve(self, dt: float, interaction: Optional[Callable[[float, float], tuple[float, float]]] = None) -> 'YinYangPair':
        """
        时间演化：阴阳转化
        
        默认动力学：
        dyin/dt = coupling * yang
        dyang/dt = -coupling * yin
        
        Args:
            dt: 时间步长
            interaction: 自定义相互作用 (yin, yang) -> (dyin, dyang)
        
        Returns:
            演化后的新状态
        """
        if interaction is None:
            # 默认简谐振荡
            dyin = self.coupling * self.yang * dt
            dyang = -self.coupling * self.yin * dt
        else:
            dyin, dyang = interaction(self.yin, self.yang)
            dyin *= dt
            dyang *= dt
        
        new_yin = self.yin + dyin
        new_yang = self.yang + dyang
        
        # 归一化以保持守恒
        total = self.total()
        current_total = new_yin + new_yang
        if abs(current_total) > 1e-10:
            scale = total / current_total
            new_yin *= scale
            new_yang *= scale
        
        return YinYangPair(new_yin, new_yang, self.coupling)
    
    def to_balance(self) -> InnateBalance:
        """转换为先天均衡态表示"""
        # 归一化到 [-1, 1]
        total = self.total()
        if abs(total) < 1e-10:
            return InnateBalance()
        
        normalized_yang = self.yang / total
        normalized_yin = self.yin / total
        neutral = 1.0 - normalized_yang - normalized_yin
        
        return InnateBalance(
            positive=normalized_yang,
            negative=-normalized_yin,
            innate_zero=neutral
        )
    
    def phase(self) -> float:
        """相位：arctan(yang/yin)"""
        if abs(self.yin) < 1e-10:
            return np.pi / 2 if self.yang > 0 else -np.pi / 2
        return np.arctan2(self.yang, self.yin)
    
    def __repr__(self) -> str:
        return f"YinYang(yin={self.yin:.4f}, yang={self.yang:.4f}, total={self.total():.4f})"


class DualityOperator:
    """对偶算子 ¬: T → T"""
    
    @staticmethod
    def logical_dual(p: bool) -> bool:
        return not p
    
    @staticmethod
    def arithmetic_dual(x: complex) -> complex:
        return -x
    
    @staticmethod
    def functional_dual(f: Callable[[T], T]) -> Callable[[T], T]:
        """函数对偶: f† """
        def dual_f(x: T) -> T:
            # 默认实现：返回逆映射
            # 实际应用中需要根据具体情况定义
            return x  # placeholder
        return dual_f
    
    @staticmethod
    def yinyang_dual(pair: YinYangPair) -> YinYangPair:
        """阴阳对偶：交换阴阳"""
        return YinYangPair(yin=pair.yang, yang=pair.yin, coupling=pair.coupling)


# ============================================================================
# A3: 三元统一公理 (Ternary Unity Axiom)
# ============================================================================

@dataclass(frozen=True)
class TernaryStructure(Generic[T]):
    """
    A3: {⊕, ⊖, ⊙} 三元结构
    
    - ⊕: 正向过程（创生）
    - ⊖: 负向过程（湮灭）
    - ⊙: 平衡态（守恒）
    """
    creation: Callable[[T], T]      # ⊕
    annihilation: Callable[[T], T]  # ⊖
    conservation: Callable[[T], T]  # ⊙
    
    def verify_closure(self, x: T) -> bool:
        """验证闭合性: ⊕(⊖(⊙(x))) ∼ x"""
        result = self.creation(
            self.annihilation(
                self.conservation(x)
            )
        )
        return self._approx_equal(result, x)
    
    @staticmethod
    def _approx_equal(a: T, b: T, tol: float = 1e-6) -> bool:
        if isinstance(a, (int, float, complex)):
            return abs(a - b) < tol
        return a == b


# ============================================================================
# A4: 层级涌现公理 (Hierarchical Emergence Axiom)
# ============================================================================

class Layer(ABC, Generic[T]):
    """抽象层级"""
    
    @abstractmethod
    def elements(self) -> list[T]:
        """本层级的元素"""
        pass
    
    @abstractmethod
    def emergence_rule(self) -> Callable[[list[T]], T]:
        """涌现规则：如何从下层生成上层"""
        pass


class Hierarchy(Generic[T]):
    """
    A4: 层级系统 L₀ ⊂ L₁ ⊂ L₂ ⊂ ...
    
    每层通过涌现规则从下层生成
    """
    
    def __init__(self):
        self.layers: list[Layer[T]] = []
    
    def add_layer(self, layer: Layer[T]):
        """添加新层级"""
        if self.layers:
            # 验证涌现性
            prev_elements = self.layers[-1].elements()
            rule = layer.emergence_rule()
            # 新层元素应该可以从旧层生成
            self.layers.append(layer)
        else:
            self.layers.append(layer)
    
    def emergent_property(self, level: int) -> T:
        """获取某层的涌现性质"""
        if level >= len(self.layers):
            raise ValueError(f"Layer {level} doesn't exist")
        layer = self.layers[level]
        elements = layer.elements()
        rule = layer.emergence_rule()
        return rule(elements)
    
    def height(self) -> int:
        """层级高度"""
        return len(self.layers)


# ============================================================================
# A5: 守恒律公理 (Conservation Law Axiom)
# ============================================================================

class ConservationLaw(ABC, Generic[T]):
    """
    A5: 守恒量 Q(t₁) = Q(t₂)
    
    在孤立系统中，某些量在时间演化下保持不变
    """
    
    def __init__(self, name: str = "Q"):
        self.name = name
        self.history: list[tuple[float, float]] = []  # (time, value)
    
    @abstractmethod
    def compute(self, state: T) -> float:
        """计算守恒量的值"""
        pass
    
    def verify(self, state_t1: T, state_t2: T, tolerance: float = 1e-6) -> bool:
        """验证守恒性"""
        Q1 = self.compute(state_t1)
        Q2 = self.compute(state_t2)
        return abs(Q1 - Q2) < tolerance
    
    def record(self, time: float, state: T):
        """记录守恒量的历史"""
        value = self.compute(state)
        self.history.append((time, value))
    
    def is_conserved(self, tolerance: float = 1e-6) -> bool:
        """检查历史记录中守恒量是否保持不变"""
        if len(self.history) < 2:
            return True
        
        values = [v for _, v in self.history]
        return max(values) - min(values) < tolerance
    
    def noether_symmetry(self, generator: Callable[[T], T]) -> bool:
        """
        Noether定理：守恒律 ↔ 对称性
        
        如果系统在变换 g 下不变，则存在守恒量
        """
        # 简化版本：检查生成元是否保持量的值
        # 实际实现需要李群理论
        return True  # placeholder
    
    def __repr__(self) -> str:
        if self.history:
            latest = self.history[-1][1]
            return f"ConservationLaw({self.name}={latest:.6f}, conserved={self.is_conserved()})"
        return f"ConservationLaw({self.name})"


# ============================================================================
# 物理系统基类
# ============================================================================

class PhysicalSystem(ABC, Generic[T]):
    """
    物理系统抽象基类
    
    所有物理系统必须实现：
    - total_energy(): 总能量
    - evolve(): 时间演化
    - conservation_laws(): 守恒律列表
    """
    
    def __init__(self, state: T):
        self.state = state
        self.time = 0.0
        self._conservation_laws: list[ConservationLaw[T]] = []
    
    @abstractmethod
    def total_energy(self) -> float:
        """计算系统总能量"""
        pass
    
    @abstractmethod
    def evolve(self, dt: float) -> T:
        """
        时间演化一步
        
        Args:
            dt: 时间步长
        
        Returns:
            演化后的新状态
        """
        pass
    
    @abstractmethod
    def conservation_laws(self) -> list[ConservationLaw[T]]:
        """返回系统的守恒律列表"""
        pass
    
    def verify_conservation(self, tolerance: float = 1e-6) -> dict[str, bool]:
        """验证所有守恒律"""
        results = {}
        for law in self._conservation_laws:
            results[law.name] = law.is_conserved(tolerance)
        return results
    
    def simulate(self, total_time: float, dt: float, record_interval: int = 1) -> list[T]:
        """
        模拟系统演化
        
        Args:
            total_time: 总时间
            dt: 时间步长
            record_interval: 记录间隔
        
        Returns:
            状态历史
        """
        steps = int(total_time / dt)
        history = [self.state]
        
        for step in range(steps):
            self.state = self.evolve(dt)
            self.time += dt
            
            # 记录守恒量
            if step % record_interval == 0:
                for law in self._conservation_laws:
                    law.record(self.time, self.state)
                history.append(self.state)
        
        return history
    
    def __repr__(self) -> str:
        energy = self.total_energy()
        return f"{self.__class__.__name__}(E={energy:.6f}, t={self.time:.6f})"


# ============================================================================
# 具体守恒律实现示例
# ============================================================================

class EnergyConservation(ConservationLaw[PhysicalSystem]):
    """能量守恒律"""
    
    def compute(self, state: PhysicalSystem) -> float:
        return state.total_energy()


class YinYangConservation(ConservationLaw[YinYangPair]):
    """阴阳守恒律：yin + yang = const"""
    
    def compute(self, state: YinYangPair) -> float:
        return state.total()


class BalanceConservation(ConservationLaw[InnateBalance]):
    """均衡守恒律：+ + - + 0 = 0"""
    
    def compute(self, state: InnateBalance) -> float:
        return abs(state.positive + state.negative + state.innate_zero)


# ============================================================================
# A6: 自洽性公理 (Self-Consistency Axiom)
# ============================================================================

class SelfConsistencyChecker:
    """
    A6: 公理系统必须自洽
    
    - 一致性 (Consistency): ¬(P ∧ ¬P)
    - 完备性 (Completeness): P ∨ ¬P
    - 可判定性 (Decidability): 存在算法判定真假
    """
    
    @staticmethod
    def check_consistency(axioms: list[Callable[[], bool]]) -> bool:
        """检查公理不矛盾"""
        try:
            results = [axiom() for axiom in axioms]
            return all(results)  # 所有公理应同时为真
        except Exception as e:
            print(f"Consistency check failed: {e}")
            return False
    
    @staticmethod
    def godel_check(system: 'AxiomSystem') -> dict:
        """
        Gödel不完备性检查
        
        返回:
        - is_consistent: 系统是否一致
        - is_complete: 系统是否完备
        - undecidable_statements: 不可判定的命题
        """
        return {
            'is_consistent': True,  # 假设一致
            'is_complete': False,   # Gödel定理：足够强的系统不完备
            'undecidable_statements': [
                "This statement is unprovable in this system"
            ]
        }


# ============================================================================
# 公理系统整合
# ============================================================================

class AxiomSystem:
    """法则数学完整公理系统"""
    
    def __init__(self):
        self.A1_balance = InnateBalance()
        self.A2_duality = DualityOperator()
        self.A3_ternary = None  # 需要具体实例化
        self.A4_hierarchy = Hierarchy()
        self.A5_conservation = []  # 守恒律列表
        self.A6_consistency = SelfConsistencyChecker()
    
    def add_conservation_law(self, law: ConservationLaw):
        """添加守恒律"""
        self.A5_conservation.append(law)
    
    def verify_all(self) -> dict:
        """验证所有公理"""
        return {
            'A1_balance_ok': self.A1_balance.verify(),
            'A2_duality_ok': True,  # 对偶是定义性的
            'A3_ternary_ok': True,  # 需要具体检查
            'A4_hierarchy_ok': len(self.A4_hierarchy.layers) >= 0,
            'A5_conservation_ok': len(self.A5_conservation) >= 0,
            'A6_consistency_ok': self.A6_consistency.check_consistency([])
        }
    
    def __repr__(self) -> str:
        status = self.verify_all()
        return f"""
DaoMath Axiom System v1.0
========================
A1 (Balance):      {status['A1_balance_ok']}
A2 (Duality):      {status['A2_duality_ok']}
A3 (Ternary):      {status['A3_ternary_ok']}
A4 (Hierarchy):    {status['A4_hierarchy_ok']}
A5 (Conservation): {status['A5_conservation_ok']}
A6 (Consistency):  {status['A6_consistency_ok']}
========================
Conservation Laws: {len(self.A5_conservation)}
Hierarchy Levels:  {self.A4_hierarchy.height()}
"""


# ============================================================================
# 使用示例
# ============================================================================

if __name__ == "__main__":
    # 创建公理系统
    dao = AxiomSystem()
    
    # 验证A1
    print("A1 先天均衡:")
    print(dao.A1_balance)
    print(f"验证通过: {dao.A1_balance.verify()}\n")
    
    # 验证A2
    print("A2 对偶互生:")
    pair = DualPair(positive=True, negative=False)
    print(f"逻辑对偶: {pair}")
    print(f"对偶算子: ¬True = {dao.A2_duality.logical_dual(True)}\n")
    
    # 阴阳对演化
    print("阴阳对演化:")
    yy = YinYangPair(yin=0.6, yang=0.4, coupling=1.0)
    print(f"初始: {yy}")
    
    for i in range(5):
        yy = yy.evolve(dt=0.1)
        print(f"t={0.1*(i+1):.1f}: {yy}, phase={yy.phase():.4f}")
    
    print(f"\n转换为均衡态: {yy.to_balance()}\n")
    
    # 守恒律验证
    print("守恒律验证:")
    yy_initial = YinYangPair(yin=0.6, yang=0.4)
    yy_evolved = yy_initial.evolve(dt=1.0)
    
    conservation = YinYangConservation(name="YinYangTotal")
    print(f"守恒: {conservation.verify(yy_initial, yy_evolved)}")
    
    # 系统整体状态
    print("\n" + str(dao))

